Аңдатпа
Abstract In the present paper, we consider the inverse problem for the polyharmonic heat equation, aiming to recover a space/time heat source F along with the temperature distribution u ( x , t ) {u(x,t)} . The problem is governed by the higher-order heat equation ( ∂ t + ( - Δ ) l ) u = F {(\partial_{t}+(-\Delta)^{l})u=F} in a finite cylindrical domain Ω × ( 0 , T ] {\Omega\times(0,T]} within the half-space ℝ d × [ 0 , + ∞ ) {\mathbb{R}^{d}\times[0,+\infty)} , where l , d ≥ 1 {l,d\geq 1} . This is done through utilizing the values of u and its normal derivatives up to order l - 1 {l-1} on a given lateral surface of the cylinder, as well as the initial value and time-averaged data (or time observation measurement) for space-dependent source and space-averaged data (energy/mass measurement) for time-dependent source. We determine that the integer k = [ d 4 l ] + 2 {k=[\frac{d}{4l}]+2} is admissible for the required degree of data regularity. The well-posedness of the classical solution to the inverse problem are established by employing the method of series expansion in terms of eigenfunctions for the Dirichlet poly-Laplacian. Weyl-type eigenvalue inequalities play a key role in our derivations.